3 21 polytope

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Up2 3 21 t0 E6.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t1 E6.svg
Rectified 321
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t2 E6.svg
Birectified 321
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 1 32 t1 E6.svg
Rectified 132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 1 32 t0 E6.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E6.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 2 31 t1 E6.svg
Rectified 231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Orthogonal projections in E6 Coxeter plane

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.[1]

Coxeter named it 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132.

These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

321 polytope[edit]

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram: CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png.

321 polytope
Type Uniform 7-polytope
Family k21 polytope
Schläfli symbol {3,3,3,32,1}
Coxeter symbol 321
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
6-faces 702 total:
126 3116-orthoplex.svg
576 {35}6-simplex t0.svg
5-faces 6048:
4032 {34}5-simplex t0.svg
2016 {34}5-simplex t0.svg
4-faces 12096 {33}4-simplex t0.svg
Cells 10080 {3,3}3-simplex t0.svg
Faces 4032 {3}2-simplex t0.svg
Edges 756
Vertices 56
Vertex figure 221 polytope
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplex.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The 1-skeleton of the 321 polytope is called a Gosset graph.

Alternate names[edit]

  • It is also called the Hess polytope for Edmund Hess who first discovered it.
  • It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure.[1]
  • E. L. Elte named it V56 (for its 56 vertices) in his 1912 listing of semiregular polytopes.[2]
  • H.S.M. Coxeter called it 321 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
  • Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)[3]

Coordinates[edit]

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:

± (-3, -3, 1, 1, 1, 1, 1, 1)

Construction[edit]

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Removing the node on the short branch leaves the 6-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 311, CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Every simplex facet touches an 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 221 polytope, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Images[edit]

Coxeter plane projections
E7 E6 / F4 B7 / A6
Up2 3 21 t0 E7.svg
[18]
Up2 3 21 t0 E6.svg
[12]
Up2 3 21 t0 A6.svg
[7x2]
A5 D7 / B6 D6 / B5
Up2 3 21 t0 A5.svg
[6]
Up2 3 21 t0 D7.svg
[12/2]
Up2 3 21 t0 D6.svg
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
Up2 3 21 t0 D5.svg
[8]
Up2 3 21 t0 D4.svg
[6]
Up2 3 21 t0 D3.svg
[4]

Related polytopes[edit]

The 321 is fifth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

k21 figures in n dimensional
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde{E}}_{8} = E8+ E10 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph Triangular prism.png 4-simplex t1.svg Demipenteract graph ortho.svg E6 graph.svg E7 graph.svg E8 graph.svg
Name −121 021 121 221 321 421 521 621

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3k1 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 {\tilde{E}}_{7}=E7+ E7++
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[[31,3,1]]
(46,080)
[32,3,1]
(2,903,040)
[33,3,1]
(∞)
[34,3,1]
(∞)
Graph 5-simplex t0.svg 6-cube t5.svg Up2 3 21 t0 E7.svg
Name 31,-1 310 311 321 331 341

Rectified 321 polytope[edit]

Rectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t1{3,3,3,32,1}
Coxeter symbol t1(321)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
6-faces 758
5-faces 44352
4-faces 70560
Cells 48384
Faces 11592
Edges 12096
Vertices 756
Vertex figure 5-demicube prism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

Alternate names[edit]

  • Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)[4]

Construction[edit]

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the node on the short branch leaves the 6-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t1311, CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 3-length branch leaves the 221, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube prism, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 2.pngCDel nodea 1.png.

Images[edit]

Coxeter plane projections
E7 E6 / F4 B7 / A6
Up2 3 21 t1 E7.svg
[18]
Up2 3 21 t1 E6.svg
[12]
Up2 3 21 t1 A6.svg
[7x2]
A5 D7 / B6 D6 / B5
Up2 3 21 t1 A5.svg
[6]
Up2 3 21 t1 D7.svg
[12/2]
Up2 3 21 t1 D6.svg
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
Up2 3 21 t1 D5.svg
[8]
Up2 3 21 t1 D4.svg
[6]
Up2 3 21 t1 D3.svg
[4]

Birectified 321 polytope[edit]

Birectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t1{3,3,3,32,1}
Coxeter symbol t1(321)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-faces 758
5-faces 12348
4-faces 68040
Cells 161280
Faces 161280
Edges 60480
Vertices 4032
Vertex figure 5-cell-triangle duoprism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

Alternate names[edit]

  • Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers)[5]

Construction[edit]

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the short branch leaves the birectified 6-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t2(311), CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 3-length branch leaves the rectified 221 polytope in its alternated form: t1(221), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-cell-triangle duoprism, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Images[edit]

Coxeter plane projections
E7 E6 / F4 B7 / A6
Up2 3 21 t2 E7.svg
[18]
Up2 3 21 t2 E6.svg
[12]
Up2 3 21 t2 A6.svg
[7x2]
A5 D7 / B6 D6 / B5
Up2 3 21 t2 A5.svg
[6]
Up2 3 21 t2 D7.svg
[12/2]
Up2 3 21 t2 D6.svg
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
Up2 3 21 t2 D5.svg
[8]
Up2 3 21 t2 D4.svg
[6]
Up2 3 21 t2 D3.svg
[4]

See also[edit]

Notes[edit]

  1. ^ a b Gosset, 1900
  2. ^ Elte, 1912
  3. ^ Klitzing, (o3o3o3o *c3o3o3x - naq)
  4. ^ Klitzing. (o3o3o3o *c3o3x3o - ranq)
  5. ^ Klitzing, (o3o3o3o *c3x3o3o - branq)

References[edit]

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen 
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 321)
  • Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq